First off, know that running is an endurance sport and not only do you have to build up your cardio level to support a sustained, raised heart rate, you have to build up the muscular endurance to bounce off of one leg for miles.
The Best Life Advice I can give when just starting is to focus equally on building core strength and stability as on building endurance and logging miles. Our core (glutes, back, abs, obliques, and all of the other muscles in our trunk) is the foundation for everything we do physically.
Three great exercises for building strength and stability in your core are:
1) Full Body Planks and Side Planks
2) Single Leg Balances
Eventually, progress to holding the balance on the ball of your foot. Start with holding for 30 seconds at a time, working on progressing to a 1-minute solid hold on the ball of your foot. Then add weight.
3) Glute Bridges
Lie on your back as if your were about to perform a crunch but instead driving your hips upwards, squeezing your booty and lowering it down for one rep. These are awesome!